Problem: What is the value of $\dfrac{d}{dx}\left(x^{^{\scriptsize\dfrac{5}{4}}}\right)$ at $x=16$ ?
Answer: Let's first find the expression for $\dfrac{d}{dx}\left(x^{^{\frac{5}{4}}}\right)$ and then evaluate it at $x=16$. The derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a fraction.) $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{^{\frac{5}{4}}}\right) \\\\ &=\dfrac{5}{4}x^{^{\frac{5}{4}-1}} \gray{\text{The power rule}} \\\\ &=\dfrac54x^{^{\frac{1}{4}}} \end{aligned}$ So we found that $\dfrac{d}{dx}\left(x^{^{\frac{5}{4}}}\right)=\dfrac54x^{^{\frac{1}{4}}}$, which can also be written as $1.25\sqrt[4]{x}$. Now let's plug ${x=16}$ : $\begin{aligned} 1.25\sqrt[4]{{16}}&=1.25\cdot 2 \\\\ &=2.5 \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\left(x^{^{\frac{5}{4}}}\right)$ at $x=16$ is $2.5$.